Solid versus fluid, and the interplay between fluctuations, correlations, and exchange in the fractional quantized Hall effect

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Abstract
In this paper, we examine a series of solid wave functions for electrons moving in two dimensions under a magnetic field, with all of them staying in the lowest Landau orbital. We discover that whereas the direct energy of the system is almost optimized by the correlated motion of electrons as in the magnetophonons, the effect of exchange is to soften the transverse branch, allowing larger fluctuations of the electrons from their lattice sites. These fluctuations are large enough to destroy the true translational long-range order of the solid and change it into mere algebraic long-range order. Thus our phonons possess a gap without violating Goldstone’s theorem. This gap is of the same order of magnitude as that exhibited by Laughlin’s fluid wave function, but the ground-state energy for the algebraic solid turns out to be slightly lower. Finally, we suggest a plausible connection between the odd-denominator rule and fractional statistics, which does not require long-range positional order and is applicable both to the solid and the fluid.